Infinite Damage & Infinite Series: Exploding Dice in Dungeons & Dragons  

Exploding D1 seems to be the best die

11:00 It would probably be better to find the chance of dealing 676 or _more_ damage, since that would also kill it. It happens that you only have to check if the 84th dice roll is a 6, 7, or 8, which triples your chances (to 3/(8^84)...)

I love the video, wonderfully explained and very informative, BUT WHAT IS UP WITH THOSE DICE WHY ARE THEY SO WONKY

15:20 This graph suggests a possible homebrew rule: When rolling (non-exploding) dice, _before_ the roll, players may _choose_ to declare their roll "exploding" and take either -1 to the roll per die (1d6 -> 1d6!-1, 2d8 -> 2d8!-2) or downgrade the die class by one rank (1d6 -> 1d4!, 2d8 -> 2d6!). This would represent taking a "risky"/"desperate" action, as the _average_ would go down slightly (for both options) in exchange for increasing the _maximum_ possible result. For example, in a situation where a character needs to roll at or above 16 (for a OHKO, (non-"d20") skill-check, or whatever) but they only have 2d6 to roll against that target...normally, that would be flatly impossible. But, if they chose to downgrade to an exploding _2d4_ instead, they'd have a _small_ chance. ...this sort of takes the idea of "automatic success on natural 20" and generalizes it to other dice (if/when a player wants to "push their luck" and take the penalty). Actually, in d20 based games, this could _replace_ the "nat-20" rule, and allow for success chances other than a flat 5% when the DC is higher than 20 + all modifiers. (You'd have to take the minus-1 option, or roll a "d18" by rolling a d20 and discarding-and-rerolling any 19s or 20s, to explode on 18s.)

Personally I homerule that healing dice explode. It makes being a healer more exciting. Attacks get crits, healing gets exploding. On average with this buff healing spells still don't outpace damage so it appears to work fine.

U can use the fact that E(X) = 1/s(1+2+…(s-1)+(s+E(X)) to solve for E(X) directly. Edit: the interesting thing is that this shows that is doesnt matter for the expected value on what number you get to roll another time

Hi, thanks for watching my SoME3 submission! I've never made an educational math video or have used Manim before, so this video is probably really amateurish. For all mathematicians and D&D rules lawyers, please comment on anything I might've gotten wrong in this video as I'm bound to have gotten something wrong. Thanks!

I played in a campaign during the D&D 3.0/3.5 era that had a variant of exploding dice. Instead of damage dice being able to explode, it was the attack roll using a d20. A natural 20 would always hit and you had roll the attack dice d20 again to confirm the bonus damage. This was a balancing mechanic that players always had a 1/20 chance of hitting an opponent regardless of how difficult it was to ordinarily hit. So a natural 20 would always hit and a natural 20 after that would always do double damage. So per our exploding dice rules, what happened if another natural 20 was rolled? Take the maximum damage and quadruple it. One more natural 20 would always insta kill your opponent regardless of their relative strength. A 1 in 160,000 to immediately slay an opponent, rare enough not to be game changing, or so we all thought. We played with a preverbal 'that guy' that embodied a chaotic character. So as 3rd level characters listening to the villainous speech of a 12th level duke oppressing the territory, that guy decided to simply throw his dagger at him from a 30 ft distance in a not so subtle means of shutting up the bad guy. Well he did got that 1/160000 luck to instakill him. Our DM had an immediate love/hate relationship with us from then on. We also had an exploding series of failures for natural 1's as well with a similar number of consecutive failures resulting in immediate player death. Our DM insisted we keep this part in our game.

Fun fact: Technically exploding dice is canon to DnD! The first level spell chaos bolt states that if the two dice you roll, both roll the same you roll them again on another target, and that applies to those dice too, so on and so forth as lucky as you can get.

As someone who knows as much about D&D as a gopher, and who knows nothing about them fancy letters in maths, this video is very interesting, I like your fancy words magic man.

You can calculate the expected value a different way. E(Dn) = (1/n)*1 + (1/n)*2 + .... (1/n)*[n + E(Dn)] this way you get a calculation that solves down to (n-1)*E(Dn) = (1 + n)*n / 2 or E(Dn) = (1+n)*n/[2*(n-1)] I checked it against your solution and the result is the same It would however be interesting to see the dispersion of the different dice (by how much on average they deviate from the average value that they roll) E([Dn-E(Dn)]^2)

Very interesting! So because the expected value of a non-exploded dice is s/2 + 1/2, we can get the the difference in damage per attack that exploding dice make by doing (s/2 + s/(s - 1)) - (s/2 + 1/2) = s/(s - 1) - 1/2. In the limit (for large s), this would mean that exploding dice do 1/2 point damage per attack more. For a d4, the difference is the largest, coming to 4/3 - 1/2 = 5/6 damage. So actually a modest improvements, certainly less powerful than +1 weapons (even setting aside the +1 to the attack roll you get for a +1 weapon). I dismissed exploding dice in D&D before, but now I think I may give my players the opportunity to gain some exploding-dice weapons at some point during my current campaign.

I don't know how common these are in 5E, but this does change the balance of 2d4 vs 1d8 and 2d6 vs 1d12. Under normal rules the average of 2d4 is 5, but the chance for 8 damage is only 6.25%, while the average for 1d8 is only 4.5, but the chance for 8 damage is 12.5%, so twice as high. If we introduce exploding dice, the average of 2d4 becomes 6.56 (rounded to 2 decimals) and the chance to deal 8 or more damage is 34.38%! The average of an exploding 1d8 is 5.13, barely above the non-exploding 2d4, and the chance to get 8 or higher is still 12.5%. Thus the only advantage the 1d8 had over 2d4 is eliminated. Though, to avoid this, one could rule that (1+n)dx only explodes when ALL dice land on the maximum. That would bring the average for 2d4 to 5.33 and the chance for 8 or more damage to 6.25%, restoring the balance with 2d4 having the higher average damage, but 1d8 having the greater chance for massive damage.

It's a nice video, thanks ! My complaint is that the music is quite too loud at the beginning and end. Also, it made me wonder how the variance behaves under the exploding dice rule.

Another interesting fact: Exploding dice improves lower dice more than higher ones. The formula for the improvement exploding gives is 1/2 + 1/(s - 1) Thus, exploding increases the average of a 1d4 by around 0.83, while 1d12 only gets an increase by around 0.59.

this is a great video! i hope you hit the algorithm soon

Really solid video, well edited. Thought this was from a much bigger channel!

***stares at the 6th level version of Shatter being presented as equivalent to the 3rd level casting of Fireball, and struggles to contain my laughter***

It's all fun and games until my tiny creature throws punches with d1s, and always rolls max damage for infinite damage.

Right before this video, I watched a video explaining the difference between infinite and arbitrarily large. In this case, the infinite series results in a potential for arbitrarily large damage, not infinite damage.

Roll and keep systems be wild with its exploding dice

Ah, Exploding dice. Closest game I have seen that uses them as standard is Savage Worlds, making use of the same dice, trying to meet or beat 4, with each level of 4 above that adding extra to the result. That said, my favorite exploding dice story comes from Deathwatch (40k Space Marine RPG, uses d10's and d%(d100). My technology specialist stayed behind to help an Imperial Guard (regular human space army guys) regiment prepare to besiege a Tau (Alien) fortress while the rest of the Marines snuck in. So while they were getting into combats and so on, I was sitting around and blessing tank turrets and ammo (standard techno-religious stuff). When I finally got the chance to shine, I rolled to hit the fortress walls, and succeeded. Then I rolled for damage...the die exploded, then again, then again, then again, then again, then again...about that time the game master told me to just stop, then described my shot hitting an ammo bunker and operating most of the outer wall of the fortress. That's the thing you have to keep in mind when describing crazy damage like that. It's less disintigrating a dragon with a dagger and more shooting the Giant in the eye with such force it penetrates through to their brain.

banger video

Loved the video. Cheers from Brazil. I used lots of graphs to decide how many dice my players would get for hit points and such, because the average of 3d6 is better than 1d18, since the minimum is 3. I play my own simplified version of D&D (15-pages long document), where very powerful characters have 30 HP (because lethality), so exploding dice are too much, but I house-ruled that exploding dice add +1 for every exploded dice. So a string of (6,6,4) actually results in 8 damage (6+2) rather than 16 (6+6+4). Considering player characters themselves start with maximum 10 HP, doing 8 damage is already a lot.

> I personally think it's a very interesting and exciting way to reward players by boosting their damage. That holds only true, as long as it is a player-only mechanic. If you do allow enemies to have exploding dice as well - then they no longer provide an advantage to players. I'd even argue that the increased damage variability creates a disadvantage for players, since players are typically supposed to be the winning party in any confrontation, and an increase in randomness always increases the chances of the underdogs to win against expectations.

Now I'm wondering about double exploding dice (so roll two more dice for every max roll you get). Does that still converge? How many more dice per max roll do we need until the expected value diverges?

Suggestion: each die explodes into s dice instead of just one die, but they retain their original values. For instance, if you roll a 6 on a d6, you then roll six more d6s and add them up. Those dice can also explode, etc. Let X be the value of a full chain for an s-sided die. E[X] = [1/s][s(s-1)/2] + [1/s][s+sE[X]], where the first term is the sum of the first s-1 numbers, each with probability 1/s, and the second term is the case where it explodes. So E[X] =(s+1)/2 + E[X]. But since s > -1, (s+1)/2 > 0, so E[X] = E[X] + k for some positive k. Therefore E[X] is not a real number. It's not hard to show that E[X] = +∞. Using the definition of expected value gives E[X] = Σ (s-1)/2 with no reference to n, in other words just the sum of a positive constant.

I think a recursion argument would be much better for explaining. It would take advantage of the fact that when you roll the maximum, you're in the same spot you were, just plus more

If a dice has n sides, then its expected value is (n+1)/2. Let E(X) be the EV of the corresponding exploding dice. Then we have E(X) = (n+1)/2 + 1/n * E(X), because there's a 1/n chance that we get to roll again. Solving this gives E(X) = n(n+1)/2(n-1) which is the same as your result. We can rewrite this as (n+1)/2 * (1 + 1/(n-1)), which shows that the extra contribution of making a dice exploding is 100/(n-1) percent.

There's a spell that sorta does explodeing dice choas bolt where the target takes 2d8 + 1d6 damage. Choose one of the d8s. The number rolled on that die determines the attack's damage type but if the d8s match no matter the number then it jumps to a new target. It's not quite exploding dice but it's always fun when jumps to another target

40 seconds in and all i can hear is the DOOOT DOOOT DOOOT from the music

yay math rocks!

alright so it does still have a higher average with larger dice BUT! what about great weapon master if you re-roll 1's and 2's and combo that along side exploding dice, does THIS change the averages in the smaller dices favor?

Exploding dice is more fun when you also explode crit fails. It's not a math problem sadly, but it adds some mechanical backing to having interesting consequences for a crit fail.

Call an exploding dice with s sides eds, and a normal dice ds. when you roll and eds, 1/s of the time you get another eds added to the roll, so if E is expected value E(eds) = E(ds) + E(eds)/s. Rearranging, and using E(ds) = (s+1)/2, you get your formula. This assumes the mean exists, but it does in this case, so no harm done. I think things get interesting when comparing say 3ed4 vs 2ed6 => E(3ed4) = 10 vs E(2ed6) = 8.4. This exacerbates the non exploding E(3d4) = 7.5 vs E(2d6) = 7 which is maybe already a bit surprising.

D&D doesn't have exploading dice, but savage worlds does. I once got 47 total damage with an exploading 2d4

"Dice" is plural. If you only have one, you don't have "a dice," you have "a die."

Part of the issue with this balance comes from weapons and spells that don’t simply roll one die. A weapon dealing a D12 is slightly worse than one dealing 2d6. With this change the 2d6 weapon becomes much better. A spell like magic missile deals 4d4 + 4 but cannot miss and is comparable to other level 1 spells that deal something like 3d8 with a chance of missing. D&d isn’t a perfectly balanced game by any means but if you adopt these rules then some classes and features are going to get much more benefit than others. Some abilities let you reroll any 1’s that you roll. That’s already a big deal in that 2d6 vs 1d12 example but when that also comes with an increase to your odds of exploding things get wild. I’d be interested in seeing the math on exploding dice when you get the option to reroll though. You’d only be able to reroll once, so if you get a 1 twice you’re stuck with it. Otherwise you’d just be rolling a 5 sided die that went from 2-6.

The only problem with implementing this house rule is that a clever (or powerbuilding) player could make use of the mechanic and some feats, races and class features to manipulate damage rolls to exploit exploding dices. It's all fun and games untill the boss fight you carefully prepared for months gets obliterated by a single guy using a calculator... Then again, it's still up to chance, so it may not be as bad as I made it out to be.

I remember the first time my DM tried introducing this rule into the game I was a player in. Due to another (Admittedly overpowered) Homebrew rule that allows for every weapon in 5E to have special abilities, I had a longbow that that as long as I could hit with disadvantage I would auto crit any target. We rolled stats and at level 1 I had a 20 in Dex. We had one combat with the rule. I crit his boss and insta killed it turn one. We were like level 5.

In my country (Czechia), most people would associate the exploding dice rule with the children's game Ludo ("Člověče, nezlob se" in Czech) where you have to roll again after roling a six, I wonder if people know that game in America

Just give me an exploding 1d1 that will always throw the highest number.

Hi there! loved the video (maybe your music was a bit too loud, great explainer nonetheless) Mathematician, DM and DnD optimizer here :P couple of things i wanted to add: i would probably write the additional damage as: (s+1)/2 + 1/2 + 1/(s-1), since (s+1)/2 is the normal damage a dice would deal, you can then see that exploding dice give an average increase of 1/2 + 1/(s-1) which averaged over a d4,d6,d8,d10 and d12 is around 2/3 additional damage. from that we can conclude... exploding dice are a really bad upgrade, worse than a straight up +1 weapon. The problem here is, that WotC seems to horribly overvalue this.

Now I get divergent vs convergent… amazing what framing can do

13:51 I need help getting this step. How can you change the series in that way?

Isn't there a value for smaller dice, at which they are more likely to roll that or higher compared to larger ones? Like, if i wanted to do x damage, and rolled a small exploding dice, wouldn't no matter the x there be a point at which the smaller dice are more likely to arrive at that damage x, compared to the larger ones?

I'm gonna throw exploding 1d1 wish me luck

Me showing up with a one sided die and some weird rules.

You could have just considered that every subsequent explosion is s times less likely than the previous one, thus you could have used the sum as n goes from 1 to infinity of [(s²+s)/2s]/(s^n). This way you could have solved for n to get [(s²+s)/2s]*(s/s-1)=(s²+s)/2(s-1) wich is equal to your result, but with less calculations. Still a very interesting video, happy to see I'm not the only fool who uses math when trying to optimize dnd.

But what if the dice explode higher or always to the same die. Like x on Dx always explode to D6 and then goes with D6.

2:53 inanimate singular they?! you use the word it to refer to bows, but dice are animate to you? Is it because the dice are the subject of "rolling" and thus are viewed as taking an active role making them take 'they' in stead of 'it'? This is very cool to me, I hope this wasn't just a one time mistake but an actual variant use :)

Does this mechanic only work when rolling ONE die? What if you're rolling an exploding 2d6?

I wonder what happens to the expected value if you "explode" on a 1

Nice video, but I'm still unconvinced of the balance of exploding dice in general. Seems like it'd be safer to roll a die of one size smaller on an explosion, that might help balance out your high-result oddity.

good video but you should work on the audio mixing, the music in the intro was way louder than your mic audio for example

The music is a bit loud compared to the voice over

I don’t think infinite series is really needed. If x is the expected value of an exploding die with n sides, then x is equal to the average roll of a regular n sided die plus a 1/n chance to roll another x. This results in x=(n+1)/2+x/n. Solving for x gives x=n(n+1)/(2(n-1)), and thus we have our expected value. For n=6 we get the expected value of 21/5, or 4.2.

the singular is DIE

This is just Acing in dead lands

The music wasn't loud enough in the beginning. i could almost hear what you were saying

Great premise, great video, great didactics. Pls dial down the music and dial up your voice on the intro for future videos, it was hardly intelligible.

Not explodey enough. Instead of adding up all those dice, multiply them. So if you roll 6,6,2 on a d6, that scores 72.

🥳❤️👍🏿

Woo hackmaster!

i though for the tarrasque it said 1/884 not 1/8^84

You sir need to get a d676 instead, after all on average with a d676 you can expect to twoshot a Tarrasque. But with a nat676 you'll oneshot it.

exploding dice are kinda used in rolemaster

The GM, of course, has the final say in all of this. The GM's job is to prevent mechanics from running away to absurd outcomes, like potential infinities, which would result in destroying the integrity of the game setting / game experience and leading to the campaign's story becoming unsatisfying. If I'm GMing, I don't actually care what the dice say; your level 1 character is not going to somehow off the main antagonist with an impossible one-shot in the 1st or 2nd session of what should be a year-long campaign. It's just not fun. It's a novelty and maybe a noteworthy story for one player to tell later, but everyone else is cheated out of the whole campaign when infinities break something that shouldn't be broken (yet) in that game world because it's an important story-driver.

just rolling again isn't much of an explosion. Would be more satisfying to me if a dice exploding meant you reroll using the next smallest dice but rolling more dice. So D20 becomes 2D12 becomes 3D10 becomes 4D8 becomes 5D6 becomes 6D4. offering a higher max outcome each time from a larger explosion of dice but it caps out and the smaller the start the smaller the finnish. If you rolled 2D6 and they both rolled 6s then you now roll 3D4s... which has the same max roll of 12. This idea can't work can it? Well damnit I've already dug my grave on this hill so I'm dying on it.

whys the music so loud lmao

2:34 hey um. i think there's something wrong with your dice. i think something's wrong

you didn't need to use infinite series

In the 1st minute, the music is too loud, and tends to drown out your voice. For what it's worth, exploding dice mechanics go back to the 80s, and D&D is a late-comer to using them. Also, the singular of 'dice' is 'die'. I don't like the standard exploding dice, because of those gaps.. Rather, it should be that rolling max means you get max-1/die plus another roll.

Exploding dice doesn't belong in D&D. Just play Shadowrun like a normal human being. They have those there.

Please don't put a doorbell thing again. It's really annoying especially since it does sound like my doorbell. I'll try to watch this video, but this is very hard. EDIT: Ok. It was cool video. I kind of expected it to not make much difference except for cases when it explodes once - where I thought it might be an edge case, but it's nice to see it calculated.